# Physics-informed deep operator network for traffic state estimation

arXiv: https://arxiv.org/abs/2508.12593 · HTML: https://mlsanity.com/paper/2508.12593/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 53 references that are in the map.
- First posted on arXiv 18 Aug 2025.

## Builds on (most cited first, 11 of 11)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Recurrent Neural Networks for Multivariate Time Series with Missing Values](https://mlsanity.com/paper/1606.01865/) (2016)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [BRITS: Bidirectional Recurrent Imputation for Time Series](https://mlsanity.com/paper/1805.10572/) (2018)
- [SAITS: Self-Attention-based Imputation for Time Series](https://mlsanity.com/paper/2202.08516/) (2022)
- [ImputeFormer: Low Rankness-Induced Transformers for Generalizable Spatiotemporal Imputation](https://mlsanity.com/paper/2312.01728/) (2023)
- [A Physics-Informed Deep Learning Paradigm for Traffic State and Fundamental Diagram Estimation](https://mlsanity.com/paper/2106.03142/) (2021)
- [Fourier neural operator for learning solutions to macroscopic traffic flow models: Application to the forward and inverse problems](https://mlsanity.com/paper/2308.07051/) (2023)
- [Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook](https://mlsanity.com/paper/2303.02063/) (2023)
- [Traffic State Estimation from Vehicle Trajectories with Anisotropic Gaussian Processes](https://mlsanity.com/paper/2303.02311/) (2023)
- [Mitigating Stop-and-Go Traffic Congestion with Operator Learning](https://mlsanity.com/paper/2411.05866/) (2024)

## Abstract (arXiv)

Traffic state estimation (TSE) fundamentally involves solving high-dimensional spatiotemporal partial differential equations (PDEs) governing traffic flow dynamics from limited, noisy measurements. While Physics-Informed Neural Networks (PINNs) enforce PDE constraints point-wise, this paper adopts a physics-informed deep operator network (PI-DeepONet) framework that reformulates TSE as an operator learning problem. Our approach trains a parameterized neural operator that maps sparse input data to the full spatiotemporal traffic state field, governed by the traffic flow conservation law. Crucially, unlike PINNs that enforce PDE constraints point-wise, PI-DeepONet integrates traffic flow conservation model and the fundamental diagram directly into the operator learning process, ensuring physical consistency while capturing congestion propagation, spatial correlations, and temporal evolution. Experiments on the NGSIM dataset demonstrate superior performance over state-of-the-art baselines. Further analysis reveals insights into optimal function generation strategies and branch network complexity. Additionally, the impact of input function generation methods and the number of functions on model performance is explored, highlighting the robustness and efficacy of proposed framework.